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On the Intraday Relation between the VIX and its Futures
Bart Frijnsa,*, Alireza Tourani-Rada and Robert I. Webbb
aDepartment of Finance, Auckland University of Technology, Auckland, New Zealand
bUniversity of Virginia, US
*Corresponding author. Department of Finance, Auckland University of Technology, Private
Bag 92006, 1142 Auckland, New Zealand. Tel. +64 9 921 9999 (ext. 5706); Fax. +64 9 921
9940; Email: [email protected].
2
On the Intraday Relation between the VIX and its Futures
Abstract
We study the intraday dynamics of the VIX and VIX futures for the period January 2, 2008 to
December 31, 2012. Considering first the results of a Vector Autoregression (VAR) using daily
data, we observe that there is some evidence of causality from the VIX futures to the VIX.
Estimating a VAR using ultra-high frequency data, we find strong evidence for bi-directional
Granger causality between the VIX and the VIX futures. Overall, this effect appears to be
stronger from the VIX futures to the VIX than the other way around. Impulse response
functions and variance decompositions confirm the dominance of the VIX futures. We further
show that the causality from the VIX futures to the VIX has been increasing over our sample
period, whereas the reverse causality has been decreasing. This suggests that the VIX futures
are become more and more important in the pricing of volatility. We further document that the
VIX futures dominate the VIX more on days with negative returns, and on days with high
values of the VIX, suggesting that those are the days when investors use VIX futures to hedge
their positions rather than trading in the S&P 500 index options.
Keywords: VIX, Futures, Vector Autoregressions, Ultra-High Frequency Data
JEL Codes: C11, C13.
3
1. Introduction
In 1993, the Chicago Board Options Exchange (CBOE) introduced the CBOE Volatility Index
(VIX).1 This index has become the leading benchmark for stock market volatility and, more
generally, for investor sentiment, due to its negative relation with the S&P 500 index.2 The
VIX has also proven to be very useful in forecasting future market volatility, where the
forecasting qualities of the VIX outperform traditional volatility measures based on realized
volatility and GARCH models (Corrado and Miller, 2005 and Carr and Wu, 2006). However,
while the VIX could be used for hedging purposes, it could not easily be traded. Theoretically,
it would be possible replicate a portfolio of the underlying options in the VIX and maintain the
30-day interpolated maturity, but the costs of doing so would be exorbitant. To expedite trading
in volatility, as well as increase hedging opportunities, the CBOE introduced futures on the
VIX on March 26, 2004. These VIX futures contracts (henceforth VXF) have become very
popular. Due to the existence of a strong negative correlation between S&P 500 index returns
and the VIX, these Futures have proven to be a far more convenient hedging tool than S&P
index options (Szado, 2010). In 2006, CBOE introduced VIX options. In addition, from
January 2009 onwards, various exchange traded product were introduced that derive their value
from VXF. These features have made the trade in VXF grow exponentially, with more than
150,000 contracts per day in 2013.
1The VIX was originally based on implied volatilities, with 30 days to expiration, of eight S&P 100 at-the-money
put and call options (Whaley, 1993). In 2003, the VIX was expanded to include options based on a broader index,
the S&P 500, reflecting a more accurate view of market volatility. The valuation model was also changed to a
model-free basis (Britten-Jones and Neuberger, 2003). 2The VIX is often referred to as the “investor fear gauge” (Whaley, 2000, 2009). Generally, when investors expect
the stock market to fall, they will buy S&P put options for portfolio insurance. By doing so, investors push up the
option prices and ultimately the level of VIX.
4
In this paper, we are particularly interested in the dynamic relation between the VIX and its
futures. With the introduction of the VIX futures, investors can hedge volatility either using
options on the S&P500 index or by taking positions in VXF. Both the demands for options in
the index and VIX futures provide useful information regarding the market’s expectation of
future volatility. An important question then arises, which is where the information on future
volatility enters the market, and which market would lead in terms of incorporating this new
information. Prior research has used daily data to (partially) address this question (see e.g.
Konstantinidi, Skiadopolous and Tzagkaraki, 2008; Konstantinidi and Skiadopolous, 2011; and
Shu and Zhang, 2012). However, inferring informational efficiency and leadership is difficult
using daily data as informational asymmetries between markets get lost in the data aggregation
process (i.e. if one market leads the other by, say, 5 minutes, then daily data will not reveal
much of this leadership). In this study, we therefore examine the dynamic relation between the
VIX and the near-term VIX futures using intraday data, where we sample at a 15-second
frequency (which is the highest frequency at which the VIX is calculated). Sampling at this
frequency eliminates all issues related to data aggregation, and allows us to get a clear picture
on the informational efficiency and leadership in the relation between the VIX and its futures.
To our knowledge, we are the first to examine the relation between the VIX and its futures
using intraday data.
We study the intraday dynamics of the VIX and VXF for the period January 2, 2008 to
December 31, 2012. Considering first the results of a Vector Autoregression (VAR) using daily
data, we observe that there is some evidence of Granger causality from the VIX Futures to the
VIX, which is in line with Shu and Zhang (2012). However, the fit of this model is poor and
the model is left with a considerably high residual correlation of approximately 0.8, suggesting
that there is a high degree of contemporaneous comovement that cannot be explained by a daily
5
VAR. Estimating a VAR using intraday data, we find that virtually all contemporaneous effects
disappear (the residual correlation is negligible), and we find strong evidence for bi-directional
Granger causality between the VIX and VXF. Overall, this effect appears to be stronger from
VXF to the VIX than the other way around. Impulse response functions and variance
decompositions confirm the dominance of the VIX Futures. In further analysis, we demonstrate
that the causality from VXF to the VIX has been increasing over our sample period, whereas
the reverse causality has been decreasing. This suggests that the VIX futures are becoming
more important in the pricing of volatility. We further document that VXF dominates the VIX
more on days with negative returns, and on days with high values of the VIX, suggesting that
those are the days when investors use VIX Futures to hedge their positions rather than trading
in the S&P 500 index options.
The remainder of this paper is structures as follows. In section 2, we review some of the
relevant literature. Section 3 describes data used in this paper and presents some summary
statistics. In section 4, we present our results. Finally, section 6 concludes.
2. Literature
Apart from the literature focusing on the volatility pricing models (e.g, Zhu and Lian, 2012;
Lu and Zhu, 2010; Brenner, Shu and Zhang, 2008, Lin, 2007, Zhang and Zu, 2006) and the
addition of a long VIX futures position to equity portfolios (Szado, 2010 and Alexander and
Korovilas, 2011) there is a limited number of empirical papers investigating the efficiency of
the VIX and VIX futures markets. As for the issue of price discovery and causality in the market
6
for volatility, there is only one study, i.e., Shu and Zhang (2012). In this section, we briefly
provide an overview of these studies on the VIX and its futures.
Konstantinidi, Skiaopoulos and Tzagkaraki (2008) and Konstantinidi and Skiaopoulos (2011)
investigate the behavior of the implied volatility indices, for the US and Europe, to assess
whether they are predictable. While the authors observe significant predictable patterns in the
futures on implied volatility indices, none of these patterns can be exploited through trading
strategies that yield economically significant profits. Hence, from an economic point of view
they cannot reject the efficiency of the volatility futures markets.
Nossman and Wilhelmson (2009) test the expectation hypothesis, using information on the
term structure of volatility, to test the efficiency of the VIX futures market. When they allow
for the existence of a volatility risk premium in their analysis, Nossman and Wilhelmsson
cannot reject futures market could predict the future VIX levels correctly.
The paper closest to our study is that of Shu and Zhang (2012). In this paper, authors explicitly
examine price discovery between the VIX and the VIX futures. They use daily prices for the
period 2004-2009. Shu and Zhang (2012) find that the VIX and the futures are indeed
cointegrated and proceed by using a Vector Error Correction model to assess the lead-lag
interaction between spot VIX and VIX futures. They find that VIX futures are informative
about spot VIX and lead the spot market in a linear error correction model. Overall, they
conclude that the VIX futures have some price discovery function.
7
3. Data and Summary Statistics
We obtain intraday data for the VIX and the futures on VIX from the Thomson Reuters Tick
History database (TRTH) maintained by SIRCA. We collect data for the period January 2, 2008
to December 31, 2012. All data are collected in tick time, with potential millisecond precision.
The CBOE computes intraday values for VIX at approximately 15-second intervals from 8.30
a.m. – 3.15 p.m. Chicago time (note that this time period reflects the normal trading hours for
the S&P 500 index options). The time interval between the calculations of the VIX is not
exactly 15 seconds, but slightly more. This implies that at the start of the day VIX may be
computed at 9:30:15.12, 9:30:30.24, etc. but during the trading day may be reported at, say,
10:30:22.54, etc.
The VIX futures were first listed on the CBOE futures exchange on March 6, 2004. The
contracts use the VIX as the underlying and use a multiplier of $1,000. The contracts trade on
CBOE Direct, the electronic trading platform of the CBOE. The minimum tick size of these
contracts is 0.01 index points. The CBOE lists up to 9 near-term serial months and five months
on the February quarterly cycle. The regular trading hours for the VXF are from 8:30 a.m. to
3:15 p.m., the same intraday period over which the VIX is computed. Trading of the contracts
terminates on the last day before the final settlement date. We collect tick-by-tick trade and
quote data for all contracts traded during our sample period. However, to construct a continuous
series, we splice together the nearest-term contracts, which are rolled over on the day when the
trading volume of the second nearest-term contract exceeds the trading volume of the nearest
8
term contract.3 We focus on the nearest-term contract as these contracts should relate closestto
the VIX. Since the contracts trade on an electronic market we collect the bid and ask quotes for
these contracts and compute the midpoint from these quotes. This frequency and sample period
provides us with a total of 1,987,254 observations for each series.
As the VIX is computed approximately every 15 seconds, we sample VIX and the VXF at a
15-second frequency.4 In addition, we exclude the first and last five minutes of the trading day,
to avoid any noise that may be due to opening and closing affecting our results.
INSERT FIGURE 1 HERE
In Figure 1, we provide a time series plot of the data. The plot shows that the VIX was relatively
low at the start of our sample period but swiftly increased and doubled from early August 2007
onwards, which can be attributed to the European Sovereign Debt Crisis. For the remainder of
the period the VIX stayed relatively high, due to the continuation of the global financial crisis,
and was decreasing only at a very slow rate. The VXF resembles the pattern of the VIX closely,
3In some cases the nearest-term contract remains the most active one until the settlement date of the contract. In
this case, we roll the contract over on the day before the last trading day, to avoid any price distortions that may
be due to final settlement of the contracts.
4Since VIX is computed at slightly more than 15 seconds, we construct 3 series, starting on the whole minute and
5 and 10 seconds after the minute. By constructing these three series, we aim to minimize the impact of stale
values of VIX that could affect our results. Note that we only report the results for the sampling interval that starts
on the whole minute. Other sampling intervals yield similar results and are available upon request.
9
but as expected the VXF is generally below the VIX when the VIX is high and is above the
VIX when the VIX is low (see also Shu and Zhang, 2012).
In Table 1, we present summary statistics on the VIX and the VXF, respectively. In the first
two columns of Table 1, we report the summary statistics for the levels of the VIX and the
VXF. Over our sample period, the VIX was on average 25.83, while the VXF was slightly
higher at 26.43. As can be seen from maximum and minimum values, the VIX has more
extreme values than the VXF. This is also reflected in the standard deviation, which is higher
for the VIX than for the VXF. Both series have positive skewness, as can be expected and have
excess kurtosis. The persistence, at the 15 second frequency, is extremely high and the first
order autocorrelation is not discernibly different from 1.00. Finally, when conducting a unit
root test on both series, we observe that we can reject the presence of a unit root for the VIX at
the 1% level, while for the VXF we can only reject the unit root at the 10% level.
INSERT TABLE 1 HERE
The last two columns of Table 1 report summary statistics for the first difference of the (log)
VIX and VXF. These first differences only include changes during the trading day, and hence
exclude the overnight change. On average, the changes in the VIX and VXF are nearly zero as
could be expected at these are ultra-high frequencies data. Again, as we observed in the levels,
ΔVIX has more extreme changes than ΔVXF as can be seen from the maximum and minimum
values, and from the standard deviations. Interestingly, the skewness of ΔVIX is negative at -
0.35, whereas the skewness of ΔVXF is nearly zero. Although both series display excess
10
kurtosis, the excess kurtosis in ΔVIX is much higher than that in ΔVXF. The first order
autocorrelation in both series is negative, at a value of -0.084 for ΔVIX and -0.118 for ΔVXF,
suggesting that there is stronger negative autocorrelation in the VIX futures. Finally, the ADF
statistics, for the difference series, suggest that we can strongly reject the presence of a unit
root in both series.
4. Results
4.1 Daily Analysis
To assess the relationship between the VIX and the VIX Futures, we start by conducting our
analysis at a daily frequency as in Shu and Zhang (2012). Since the summary statistics show
that there is weak evidence of a unit root in the VIX futures, we compute first differences of
the log of the volatility series. We then estimate the following VAR:
122221
111111
)()(
)()(
tttt
tttt
VXFLVIXLVXF
VXFLVIXLVIX
, (1)
where ϕ1(L), φ1(L), ϕ2(L), and φ2(L), are polynomials in the lag operator of identical length.
Daily data suggests an optimal lag length of one using the Schwartz Information Criterion
(SIC). Hence, we estimate Equation (1) as a VAR(1).
11
In Table 2, we report the regression results for the VAR(1) as well as Newey-West adjusted t-
statistics in parentheses. For ΔVIXt, we find a negative and significant coefficient on ΔVIXt-1
suggesting that there is negative autocorrelation in changes in the VIX at the daily frequency.
We also find a positive coefficient for ΔVXFt-1, significant at the 5% level, suggesting that at
the daily frequency the VIX futures have some predictive value for the changes in the VIX.
For ΔVXFt, we find that there is no statistical evidence for predictability of the changes in the
VIX futures, neither lagged changes in the VIX or the VXF can be used to predict these
changes. Although we find some evidence of predictability for ΔVIX, we note that the R2’s of
the regressions are quite low at 1.82% and 0.18% for ΔVIXt and ΔVXFt, respectively. The
contemporaneous correlation between the residuals in the VAR is quite high at 0.813,
suggesting a strong correlation between the series. Granger causality tests, reported in Panel B,
confirm the findings of the coefficients, i.e., there is Granger Causality from ΔVXF to ΔVIX,
but not the reverse. Overall, the results of our daily analysis are in line with Shu and Zhang
(2012).
4.2 Intraday Analysis
The daily analysis reveals some evidence for predictability of changes in the VIX. However,
this analysis also revealed a very high contemporaneous correlation between the changes in the
VIX and the VXF. Part of this contemporaneous correlation may be due to data aggregation.
Sampling at higher frequencies may reduce the contemporaneous correlation and reveal more
lead-lag dynamics. We therefore estimate the VAR in Equation (1) using intraday ultra-high-
frequency data sampling at a 15 second frequency. Instead of estimating Equation (1) as one
12
big VAR using 1,987,254 observations, we estimate the model every day in the sample, so that
we obtain a daily series of coefficients and statistics.
First, we determine the optimal lag length of the VAR by computing the SIC for 1 lag up to 10
lags every day. Then, we compute the average SIC over all days in the sample. We find that
the average SIC is lowest for a VAR with three lags. Hence, we estimate all coefficients and
compute all statistics based on daily models using three lags.
In Panel A of Table 3, we report the results for the intraday VAR(3) model. We report
coefficients and indicate significance using asterisks (based on Newey-West corrected standard
errors obtained from the time series of coefficients). In brackets, we report the 2.5% and 97.5%
percentile values from the time series of coefficients.
When we consider the dynamics of ΔVIX, we find evidence of some negative autocorrelation,
with the first and second lag significant at the 5% level and values of -0.047 and -0.020,
respectively. The coefficient for the third autoregressive lag is positive at 0.005 and although
very small, it is significant at the 10% level. For the coefficients on the lagged values of ΔVXF,
we find that all three coefficients are positive and significant at the 1% level, with values of
0.172, 0.119, and 0.058, for lags one, two and three, respectively. This provides evidence that
there is some predictability for the changes in the VIX based on lagged changes in the VXF.
The R2 of this regression, 9.26%, is considerably higher than that for the regression using daily
data, suggesting that much more of the variation in the intraday changes in the VIX can be
explained by past information. For the changes in the VXF, we note that lagged values of ΔVIX
13
have a positive and significant effect on changes in the VXF for all three lags. This finding
suggests that there is some predictability of changes in the VXF based on lagged values of
ΔVIX. This contrasts the findings that we observed for the daily estimation. We also find
significant evidence for negative autocorrelation in the changes in the VIX futures at this
frequency, with all three lags yielding negative and significant coefficients. Again, compared
with the daily VAR, the R2 of the intraday VAR is considerably higher at 4.12%, but still lower
for the regression for ΔVIX. At this high level of data frequency, we also observe that the
contemporaneous correlation in the changes in VIX and VXF is close to zero (on average
0.0019, with 2.5% and 97.5% percentile values at -0.0001 and 0.0058, respectively). This
finding indicates the presence of high positive contemporaneous correlation between daily
changes in VIX and VXF is driven by data aggregation.
INSERT TABLE 3 HERE
In Panel B of Table 3, we report the results for Granger causality tests, where we report the
average Granger causality statistic over the sample period and report the percentages of
significant Granger causality statistics at conventional significance levels. When we consider
the Granger causality from ΔVXF to ΔVIX, we find that the average coefficient is equal to
102.80, thus giving very strong statistical evidence for a causal relation running from ΔVXF to
ΔVIX (note that the 1% critical value for this statistic is 11.30). When we consider the
percentage of days that we find a significant effect of ΔVXF on ΔVIX, we find that between
92.37% (85.45%) of the days there is significant causal effect measured at the 10% (1%)
14
significance level. When we investigate the causal effect of ΔVIX on ΔVXF, we find an
average test statistic of 14.31, which, although lower than the causality statistic of ΔVXF on
ΔVIX, is still highly significant. Next, we compute the percentage of days that there is a causal
effect of ΔVIX on ΔVXF. It is observed that on 63.99% (42.05%) of the days there is significant
evidence for causality running from ΔVIX to ΔVXF measured at the 10% (1%) level. Overall,
the Granger Causality tests reveal that there is significant evidence for reverse causality, but
the effect appears to be stronger running from ΔVXF to ΔVIX than vice versa.
Given that the contemporaneous correlation between ΔVIX and ΔVXF is nearly zero, we can
easily estimate impulse response functions and compute variance decomposition assuming that
both series are contemporaneously uncorrelated.5 In Figure 2, we plot the impulse response
functions for 10 steps ahead, where we apply a unit shock to the residuals of each series. Panels
A and B of Figure 2 show the responses to a unit shock in ΔVIX. As can be seen, this shock
has some impact on changes in the VIX, but dies out after about four periods. Panel B shows
that this shock leads to a small increase in the VXF after which it decreases and again dies out
after about four periods. Overall, a unit shock to ΔVIX does not lead to a change of more than
0.1 (in absolute terms) in the VXF. Panels C and D show the responses of a unit shock in
ΔVXF. Considering Panel D first, we note that a unit shocks to ΔVXF leads to a drop in the
VXF after about two periods, and dies out after about 5 periods. The response of ΔVIX to a
shock in ΔVXF (Panel C), shows a positive reaction in ΔVIX after two periods and dies out
again after about 5 periods. Overall, the impulse response analysis shows that there is
5Note that this additional analysis is difficult to conduct in a meaningful way using daily data, due to the high
contemporaneous correlations observed in daily data.
15
bidirectional spillover between ΔVIX and ΔVXF, however, the response in the VIX to a shock
in the VXF is substantially larger than the response of the VXF to a shock in the VIX.
The last rows of Table 3 report the Variance Decomposition, where we decompose the variance
of ΔVIX (ΔVXF) that is due to either ΔVIX or ΔVXF. When we decompose the variance of
ΔVIX, we find that 94.53% of the variance comes from ΔVIX, whereas 5.47% originates from
changes in the VXF. Vice versa, the variance of changes in the VXF is for 3.54% due to
changes in the VIX and the remaining 96.46% is due to its own changes. In line with the
Granger causality tests and the impulse response functions, this result suggests that the changes
in the VXF have a greater influence on the changes in the VIX then the other way around.
4.3 Time-variation in Granger Causality
The next question we address is whether we observe any time variation in the intraday causal
relation between ΔVIX and ΔVXF. In Table 4, we report Granger causality statistics per year
during our sample period. The first column of Table 4 reports the causality statistics from
ΔVXF to ΔVIX. We note that since 2008, there is an upward trend in the causality statistics,
indicating that causality from ΔVXF to ΔVIX has becoming stronger. However, we also
observe a slight drop off in the causality statistic in the last year in the sample in 2012. For the
causality in the opposite direction, we observe a downward trend in the statistic going from an
16
average of 25.896 in 2008 to 7.151 in 2012, which is just below the 5% significance level.
Hence the causal effect of ΔVIX on ΔVXF seems to have died off over time.
INSERT TABLE 4 HERE
In the next two columns of Table 4, we report the average statistics for the variance
decomposition per year over our sample period. The first of these columns reports the
percentage of variance of ΔVIX that is attributable to ΔVXF. Again, we note an upward trend
over time going from 2.62% in 2008 to 10.04% in 2011, after which it declines to 5.86% in
2012. The percentage of variance of ΔVXF attributable to ΔVIX shows a less clear picture,
which starts at 4.66% in 2008 and declines to 3.06% in 2012.
To provide a visual representation of the time variation in the Granger Causality between ΔVIX
and ΔVXF, we plot the 10-day moving average of the log of the ratio of the Granger causality
statistics in Figure 3, i.e. log(GCΔVXF/GCΔVIX), where GCΔVXF is the Granger causality statistic
of causality from ΔVXF to ΔVIX, and GCΔVIX is the Granger causality statistic of causality
from ΔVIX to ΔVXF. From this plot, we observe that there is an increase in this ratio. At the
start of the sample period the ratio is less than zero, suggesting that the causality from ΔVIX
to ΔVXF is stronger. However, this ratio quickly becomes positive. Overall, there is an upward
trend in this ratio.
INSERT FIGURE 3 HERE
17
In order to explain what drives the changes and increase in this ratio, we conduct the following
regression analysis, i.e.
t
t
ttt
tVIX
VXF
OptVol
FutVolVIXSPRtrend
GC
GC
log)log(_log 321 , (2)
where
tVIX
VXF
GC
GC
log is the daily ratio of the Granger causality statistics, trendt captures the
time trend that we observed in the ratio, R_SPt is the return on the S&P 500 index on day t,
log(VIX)t is the log of the VIX on day t, and t
OptVol
FutVol
log is the log of the ratio of daily
volume traded in the near-term VIX futures relative to the daily volume traded in the near-term
S&P 500 index options on day t.6
In Table 5, we report the results for Equation (2), where we include variables step-by-step. We
report all coefficients and Newey-West corrected t-statistics in parentheses. First, we estimate
the regression only including the time trend. The results, in the first column of Table 5, show
that this time trend is highly significant, and this regression produces an adjusted R2 of 28.43%.
6Note that we use a detrended version of this ratio as there is a strong positive upward trend in this variable. We
have also conducted the analysis with the ratio of daily volume traded in the near-term VIX futures relative to the
volume traded in the near-term S&P500 index put options. The results for this analysis are nearly identical to the
ones reported in this paper.
18
This provides clear evidence of an upward trend in the informativeness of the VIX futures over
the VIX. In the second column, we add the returns on the S&P500 index. The coefficient on
these returns is negative and significant at the 1% level. This suggests that VIX futures are
more informative on days when returns are negative and could be due to the increased hedging
using VIX futures on days with negative returns. The adjusted R2 increases slightly relative to
the model that only includes the time trend to 29.28%.
INSERT TABLE 5 HERE
Next, we include the log of the VIX. We find that the coefficient for this term is positive and
significant at the 1% level. Hence this suggests that when the VIX is high, the informativeness
of the VXF relative to the VIX increases. Again, this could be due to the increased hedging
activity in using VIX futures, when uncertainty (VIX) in the market is high. The adjusted R2
of this regression is 31.40%, suggesting that the VIX is more informative for the causality ratio
than the returns on the S&P 500. In the next column, we include the ratio of volume traded in
the VIX futures relative to the volume traded in the S&P 500 options. One reason why we
could expect an increase in the informativeness of the VXF is because more people are using
the VIX futures to hedge their positions than the options on the S&P500. We observe that the
ratio is insignificant in this regression, showing that an increase in activity in the VIX futures
relative to the options does not explain the ratio of causality. Finally, we estimate a regression
where we include both the returns on the S&P 500 and log of the VIX. Prior literature has
shown that there is a strong and negative relation between these two variables (e.g. Whaley,
2000) and hence we need to include both in a single regression to determine whether the
19
negative relation between the returns on the S&P500 and the relative informativeness of the
VXF is driven by the VIX, and vice versa. When we include both variables, we observe that
both maintain their sign and significance, suggesting that both returns on the S&P500 and the
VIX are informative for the causality between the VIX and its futures.
5. Conclusion
In this paper, we examine the intraday dynamics of the VIX and VXF for the period January
2, 2008 to December 31, 2012. In line with Shu and Zhang (2012), we observe some evidence
of Granger causality from the VXF to the VIX at a daily level. However, at the intraday level,
we find strong evidence for bi-directional Granger causality between the VIX and the VXF.
Overall, this effect appears to be stronger from the VXF to the VIX than the other way around,
which is confirmed by impulse response functions and variance decompositions. We document
that the causality from the VXF to the VIX has been increasing over our sample period, whereas
the reverse causality has been decreasing. This suggests that the VIX futures are become more
and more important in the pricing of volatility. We further find that the VIX futures dominate
the VIX more on days with negative returns, and on days with high values of the VIX,
suggesting that those are the days when investors use VIX futures to hedge their positions
rather than trading in the S&P 500 index options. Overall, our results suggest that the VIX
futures are informationally dominant over the VIX in reflecting future volatility.
20
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23
Table 1. Descriptive Statistics
Levels Differences
VIX VXF ΔVIX ΔVXF
Mean 25.83 26.43 -4.14e-06 -8.22e-07
Median 22.60 24.11 0.00 0.00
Max. 88.05 69.26 0.197 0.054
Min. 13.30 15.08 -0.212 -0.049
Standard
Deviation
11.03 9.17 0.00130 0.000996
Skewness 1.944 1.647 -0.353 0.0415
Kurtosis 7.218 5.687 1785 33.91
ρ1 1.00 1.00 -0.084 -0.118
ADF -3.64*** -2.78* -121.27*** -
450.65***
24
Table 2. Daily VAR Analysis
Panel A: Estimation Results
ΔVIXt ΔVXFt
C -0.0001
(-0.10)
-0.0002
(-0.18)
ΔVIXt-1 -0.206***
(-3.35)
-0.050
(-1.52)
ΔVXFt-1 0.149**
(2.05)
0.047
(0.94)
R2 0.0182 0.0018
Panel B: Granger Causality Tests
Causality from ΔVXF to
ΔVIX
Causality from ΔVIX to
ΔVXF
4.84** 2.12
25
Table 3. Intraday VAR Analysis
Panel A: Estimation Results
ΔVIX ΔVXF
constant -4.80e-06***
[-6.11e-05,5.82e-05]
-1.07e-06
[-4.60e-05, 5.24e-05]
ϕ11 -0.047***
[-0.486, 0.169]
0.056***
[-0.036, 0.174]
ϕ12 -0.020***
[-0.295, 0.124]
0.033***
[-0.053, 0.127]
ϕ13 0.005*
[-0.154, 0.128]
0.017***
[-0.074, 0.096]
φ11 0.172***
[0.015, 0.503]
-0.142***
[-0.289, 0.003]
φ12 0.119***
[-0.010, 0.391]
-0.066***
[-0.166, 0.028]
φ13 0.058***
[-0.033, 0.203]
-0.034***
[-0.108, 0.036]
R2 0.0926
[0.0057,0.3147]
0.0412
[0.0075,0.0954]
Panel B: Granger Causality and Variance Decomposition
Causality from ΔVXF to ΔVIX Causality from ΔVIX to ΔVXF
Mean 102.80 14.31
Perc. exceeding 10% Crit. Val. 92.37% 63.99%
Perc. exceeding 5% Crit. Val. 90.38% 57.15%
Perc. exceeding 1% Crit. Val. 85.45% 42.05%
Variance Decomposition ΔVIX ΔVXF
due to ΔVIX 94.53% 3.54%
due to ΔVXF 5.47% 96.46%
26
Table 4. Causality and Variance Decomposition by Year
Causality from
ΔVXF to
ΔVIX
Causality from
ΔVIX to ΔVXF VD ΔVIX due
to ΔVXF
VD ΔVXF due
to ΔVIX
2008 39.722 25.896 2.62% 4.66%
2009 51.552 17.622 2.77% 2.80%
2010 108.320 12.903 6.08% 3.49%
2011 220.061 7.870 10.04% 3.68%
2012 94.556 7.151 5.86% 3.06%
27
Table 5. Regression Results for the Causality Ratios
tVIX
VXF
GC
GC
log
tVIX
VXF
GC
GC
log
tVIX
VXF
GC
GC
log
tVIX
VXF
GC
GC
log
tVIX
VXF
GC
GC
log
α 0.0366
(0.260)
0.0258
(0.18)
-3.592***
(-3.43)
0.00366
(0.259)
-3.398***
(-3.21)
trendt 0.0027***
(12.21)
0.0027***
(12.28)
0.0032***
(12.94)
0.0027***
(12.16)
0.0032***
(12.82)
R_SPt -0.105***
(-4.10)
-0.081***
(-03.18)
log(VIX)t 1.041***
(3.37)
0.983***
(3.155)
tOptVol
FutVol
log
0.0458
(0.43)
R2(adj) 0.2843 0.2928 0.3140 0.2839 0.3187
28
Figure 1. Time Series Plot of VIX and VXF
VIX
VXF
29
Figure 2. Impulse Response Functions
30
Figure 3. Moving Average of the Log Causality Ratio